Let $\mathfrak{g}$ be a semisimple Lie algebra and $M, N$ be two modules of $\mathfrak{g}$. Is it true that $M \otimes N \cong N \otimes M$? If $\mathfrak{g}$ is replaced by other algebras, $M \otimes N \cong N \otimes M$ is also true? What conditions do we need in order to have $M \otimes N \cong N \otimes M$? Thank you very much.

1 Answer
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There are two things going on here. The first is that you are taking the tensor product of two vector spaces. The second is that you have Lie algebra structures.

For any two vector spaces, we have an isomorphism $V\otimes W \to W\otimes V$ given by $v\otimes w \mapsto w\otimes v$. We can easily get that this exists and is an isomorphism by using the universal property: a bilinear map from $V\times W$ is easily turned into a binlinear map from $W\times V$ by defining $\widetilde{f}(w,v)=f(v,w)$. If you have not seen this before, you should try to work out the details.

The issue that remains is whether this isomorphism of vector spaces is compatible with the Lie algebra structure. Let $\tau(v\otimes w)=w\otimes v$. Then we must show that for each $g\in \mathfrak g$, $\tau(g.v\otimes w)=g.\tau(v\otimes w)$. Note that by linearity, it suffices to check the statement on simple tensors. This is a straightforward calculation, using only the definition of the $\mathfrak g$-module structure on the tensor product.

More generally, what is going on is that the universal enveloping algebra $\mathcal U\mathfrak g$ is a Hopf algebra with comultiplication $\Delta:\mathcal U\mathfrak g\to \mathcal U\mathfrak g\otimes \mathcal U\mathfrak g$ defined on generators by the formula $\Delta(g)=g\otimes 1 + 1 \otimes g$ and extended by the condition that $\Delta$ is a map of algebras (exercise: verify that this map is well defined).

The Hopf algebra structure is what allows us to define an action on $V\otimes W$. Because a $\mathfrak g$-module is the same thing as a $\mathcal U\mathfrak g$-module, $V\otimes W$ is a priori only a module over $\mathcal U\mathfrak g\otimes \mathcal U\mathfrak g$. To get the structure of a $\mathcal U\mathfrak g$-module, we must pull back along $\Delta$. The module structure comes from the composite
$$\mathcal U\mathfrak g\to \mathcal U\mathfrak g\otimes \mathcal U\mathfrak g \to \operatorname{End}(V)\otimes \operatorname{End}(W) \to \operatorname{End}(V\otimes W).$$

where all the maps are maps of algebras. The fact that $\tau$ is a map of $\mathfrak g$-modules comes from the easy to verify fact that $\tau \Delta = \Delta \tau$ (where $\tau$ in this context is the swapping map on $\mathcal U\mathfrak g \otimes \mathcal U\mathfrak g$).

In general, the category of modules over a cocommutative Hopf algebra, or comodules over a commutative Hopf algebra will have a commutative tensor product induced from the tensor product of vector spaces. More generally, one can study quantum groups, where the commutivity or cocommutivity is weakened so that instead of the category of representations being symmetric monoidal, it is braided monoidal. In this way, understanding quantum groups allows one to better understand representations of the braid group.